Problem 1: Let X be a topological space. Let C be the category whose objects are the open subsets of X and whose morphisms are the containment maps. Given any family of open subsets Ui indexed by a set I, determine if C contains a categorical product of these objects. If so, give a construction of the product in this category.
Problem 2: Let A and B be objects in a category C. Prove that an isomorphism f from A to B is both a monomorphism and an epimorphism.
Problem 3: Do coproducts exist for arbitrary families of objects in the category in Problem 1? If so, give a construction of the coproduct.